Simplify the following expression and state the condition under which the simplification is valid. $r = \dfrac{a^2 - 81}{a + 9}$
First factor the polynomial in the numerator. The numerator is in the form ${a^2} - {b^2}$ , which is a difference of two squares so we can factor it as $({a} + {b})({a} - {b})$ $ a = a$ $ b = \sqrt{81} = 9$ So we can rewrite the expression as: $r = \dfrac{({a} + {9})({a} {-9})} {a + 9} $ We can divide the numerator and denominator by $(a + 9)$ on condition that $a \neq -9$ Therefore $r = a - 9; a \neq -9$